Agriculture Two Weeks
/ GeometryLesson Plan
Teacher:
8th Grade Math Teacher / Grade:
8th Grade
Lesson Title:
Congruence and Right Triangles: Their Properties and How They Lend to Trigonometric Ratios and Real World Relevance
STRANDS
Similarity, Right Triangles, and Trigonometry
Congruence
LESSON OVERVIEW / Summary of the task, challenge, investigation, career-related scenario, problem, or community link.
The unit will start with an investigation of congruence and the real world relevance of using congruence in order to be time and cost efficient. Students will then investigate how different transformations affect the congruency of triangles. Students will also learn the minimum information needed in order to recreate a triangle that is congruent to the original. This will lead into student investigating right triangles and how their similarities make way into the trigonometric ratios that allow humans to solve large-scale measurement problems. Social Studies will be examining agricultural practices used in the past, how they have changed over time, and how agriculture has shaped the United States. Language Arts will be exploring agriculture through literature as well as practicing technical writing at the conclusion of the lab. As well as examining physical and chemical changes students will also develop a working definition for matter, gain an understanding of density, and investigate states of matter. This will culminate in a raised bed vegetable garden project that will integrate the finding from all subjects. Ben Hunter, with the Agriculture Extension Office, will come during the final 2 project days to assist and educate students on the construction and maintenance of the raised bed vegetable gardens.
MOTIVATOR / Hook for the week unit or supplemental resources used throughout the week. (PBL scenarios, video clips, websites, literature)
Day 1 -"Modular Efficiency":
This motivator will utilize the following video clip – “Modular Efficiency” (Appendix A). The students will then discuss the pros and cons of modular construction. What type of figures are these modules used to build (congruent figures), and how does having modules (congruent figures) make a company more efficient? The teacher will lead the discussion of how modular construction is utilized in STEM fields in order to be cost and time efficient.
Day 4 -"Distance to the Stars":
This motivator will utilize the following video clip – “Distance to the Stars” (Appendix F). The students will then discuss how trigonometry is used in today’s society after watching the informative video. Things will be brought up such as GPS and satellite location. The students will then debate Other types of STEM careers and technology that involve the use of triangles and trigonometry. Teacher will lead the discussion by offering how trigonometry is a leading component in structural analysis, as a measuring device for all things that are too large to physically measure, and sailing.
DAY /
Objectives
(I can….) /Materials & Resources
/Instructional Procedures
/ DifferentiatedInstruction /
Assessment
1 / I can use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. / “Modular Efficiency”(Appendix A)
“Verifying Triangle Congruence”
(Appendix B)
Ruler (or straight edge)
Calculator
“Need More Support” Verifying Triangle Congruence Task
(Appendix C)
“Need More Challenge” Triangle Congruence Task
(Appendix D) / Essential Question(s):
How can I use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent? / Remediation:
Peer Tutoring
Heterogeneous Grouping
“Need More Support” Triangle Congruence Task. This gives the graphed figure with the angles and side measurements already completed. It also gives the students the different types of transformations they are investigating.
Enrichment:
Peer Tutoring
Heterogeneous Grouping
“Need More Challenge” Triangle Congruence Task. This deepens students understanding of congruence by questioning what transformations do not conserve congruence, but do conserve similarity within the transformed figures. / Formative Assessment:
Teacher Observations
Performance Assessment:
Exit Ticket
Summative Assessment:
Verifying Triangle Congruence Task graphs and Results
Set:
Teacher will begin by showing the “Modular Efficiency” video clip, The students will then discuss the pros and cons of modular construction. What type of figures are these modules used to build (congruent figures), and how does having modules (congruent figures) make a company more efficient? The teacher will lead the discussion of how modular construction is utilized in STEM fields in order to be cost and time efficient. The teacher will then ask the students how they may prove that these figures are congruent (sides, angles.)
Teaching Strategy:
1. Place students in pairs and hand out “Verifying Triangle Congruence” Task. Also have available the “Need More Support” Triangle Congruence Task and “Need More Challenge” Triangle Congruence Task for students in need of differentiated instruction.
2. Have the students work together to transform the figure, and determine if the new figure is congruent to the original figure.
3. Come back together as a class and conduct a class discussion using the following discussion questions:
· “How does a reflection, rotation, or translation affect the corresponding parts of a transformed figure?”
· “Is there a transformation that does not create a congruent figure?”
“Using your prior knowledge of angles, side lengths, and triangles is there an alternative approach to verifying the triangles are congruent besides showing that all of the corresponding sides lengths are congruent and all of the corresponding angles are congruent.”
Summarizing Strategy:
As an exit ticket, have students summarize their findings. What needs to be true in order for triangles to maintain congruency (sides and angles)? What is the only transformation that does not maintain congruency?
2 / I can explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. / Construction paper
“Bulletin Board Congruence” Task (Appendix E)
Rulers
Protractor
Markers / Essential Question(s):
How can I explain the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions? / Remediation:
Peer Tutoring
Heterogeneous Grouping
Enrichment:
Peer Tutoring
Heterogeneous Grouping / Formative Assessment:
Teacher observations of methods used to create congruent triangles
Performance Assessment:
Ending discussion of methods used by students
Summative Assessment:
Completed Bulletin Boards
Exit Ticket
Set:
Begin by asking students to jot down what it means for two or more triangles to be congruent to one another. Ask them to think about and write down how they would go about creating congruent triangles. Come together as a group and discuss answers from the students
Teaching Strategy:
1. Divide the class into groups of 2-3. Provide the students with a copy of the “Bulletin Board Congruence” Task and a large piece of construction paper. The paper will be used as the size of the bulletin board. Allow students a few minutes to read over task and make any additional notes or jot down any questions.
2. Allow students time to work on the task and create solutions. Walk around and observe the students. Ask them about their techniques and how they are meeting the goal of the task.
3. Allow the class to complete a gallery walk to see the different solutions from around the classroom.
Summarizing Strategy:
As an exit ticket, have students summarize their findings. What is the smallest number of measurements needed (sides and/or angles) to recreate congruent triangles?
3
Project Day – See Unit Plan
Feeding America: Exploring Raised Bed Gardening – Research and Planning
4 / I can understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. / Printer Paper
“Distance to the Stars” Video Clip (Appendix F)
“Graphic Chart” (Appendix G)
“Trigonometric Table Spread Sheet.”
(Appendix H)
Graphing Calculator
Ruler
Protractor
Ruler
“Pre-designed Right Triangle”
(Appendix I)
“Need More Challenge” Task
(Appendix J) / Essential Question(s):
How can I understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles? / Remediation:
Peer Tutoring
Give students the “Pre-designed Right Triangle”
with the 30o angle, hypotenuse, adjacent side and opposite side clearly identified.
Enrichment:
Peer Tutoring
Give students the option of the “Need More Challenge” Task in which students deepen their understanding of the relationship between the acute angle, length of sides, and their trigonometric properties. / Formative Assessment:
Teacher observations of student conclusions
Performance Assessment:
Discussions throughout task
Summative Assessment:
Completed “Graphic Chart”
Triangle Designs
Exit Ticket
Set:
Begin the class by showing the video clip: “Distance to the Stars.” The students will then discuss how trigonometry is used in today’s society after watching the informative video. Things will be brought up such as GPS and satellite location. The students will then debate Other types of STEM careers and technology that involve the use of triangles and trigonometry. Teacher will lead the discussion by offering how trigonometry is a leading component in structural analysis, as a measuring device for all things that are too large to physically measure, and sailing.
Teaching Strategy:
1. Give each student a sheet of unlined paper a protractor and a ruler. Instruct students to draw, as accurately as possible, a right triangle with one of its acute angles measuring 30o. Ask students, in pairs, to compare triangles. Ask students to share their observations. (Students should recognize that, while some of the triangles may be approximately congruent, all the triangles in the room are approximately similar.)
2. Have students form trios by finding two partners that created triangles that are “different” than theirs. Ask, “What are the differences that made you choose your group members? What is still the same about your triangles?”
3. Have groups measure, as accurately as possible, the lengths of all three sides of their triangles and find the measure of the unknown angle. (They may find the angle by measuring, but some may use the fact that the sum of the measures of the angles of a triangle is 180o to find that the remaining angle measures 60o.) Explain to students a system of identifying the sides of their triangles. (One side is the “hypotenuse” and will be denoted as “H.” The hypotenuse will be a previously learned concept. It may be necessary to review it in the context of the Pythagorean theorem. The legs can now be identified relative to one of the acute angles called the “reference angle.” The leg across from the reference angle is called the “opposite leg,” which will be denoted as “O,” and the leg which makes up one side of the reference angle is called the “adjacent leg” which will be denoted as “A.”) Instruct students to use the 30o angle as the reference angle. Have each group choose one pair of sides and create a ratio of those sides for each triangle in the group and find a decimal approximation of the ratio. (Groups should find that the ratio they find is equal for all three triangles.) Have groups report their results. Record the results on a board or projector (using the graphic organizer). Give each group the “Graphic Chart” to record the results for any of the ratios found by the class. If all six ratios have not been chosen, have the class find the remaining ratios in groups and report them to complete the table. Have groups add these labels to three of the ratios on the chart: O/H = sine 30o, A/H = cosine 30o, O/A = tangent 30o. Point out abbreviations “sin, cos, tan.” (Anticipate the possibility of questions as to why the other three ratios, cosecant or csc and secant or sec and cotangent or cot, are not being named at this point. In this case, point out that these ratios are merely reciprocals of the first three and not necessary for relating the sides at this point. Feel free to identify them by name to satisfy curiosity.) Ask,
· In any right triangle with a 30o angle, what do we now know about the ratio of the leg opposite the 30o reference angle to the hypotenuse (O/H)? (It is always the same and equal to .5.)
· In any right triangle with a 30o angle, what do we now know about the ratio of the leg adjacent to the 30o reference angle to the hypotenuse (A/H)? (It is always the same and equal to approximately .8660.)
· In any right triangle with a 30o angle, what do we now know about the ratio of the leg opposite to the 30o reference angle to the leg adjacent to the 30o reference angle (O/A)? (It is always the same and equal to approximately .5774.)
· What is the measure of the other acute angle? What would happen to the trig ratios if we used this angle as the reference angle?
4. Ask, “What would happen to the trig ratios if we used a different angle? What, more specifically, do you think would happen to the sin and cos of the angle if we reshaped the triangle so that the reference angle was changed from 30o angle to 40o? Be prepared to explain why you think this will be true.” Have students discuss the accuracy of the answers. Elicit responses from groups who found specific answers gleaned from ratios. Note these answers as with 30o before. Look for or make a suggestion that there are other ways than construction of a triangle to find the ratios.
5. Make sure each group has a scientific or graphing calculator. (One calculator per student is preferred.) Ask them to explore and find the sin30o, cos 30o, sin 40o, and cos 40o on their calculators. Tell them they will know they have found it correctly, without using a constructed right triangle, when they get the same approximate answers as discovered earlier. Ask them to find tan of the two reference angles as well.
6. Give each student a copy of the “Trigonometric Table Spread Sheet.” Ask students to find the sin, cos, and tan of 30o and 40o on the table and compare the answer to that on the calculator. (Note that the answers are approximate and that the decimals for the ratios non-terminating and non-repeating. Ask, “What generally happens to the sine as the reference angle gets bigger? What happens to the cosine and tangent?” Other than, looking at answers on the table or calculator, why do you think this is true?